Enumeration of finite monomial orderings and combinatorics of universal Gröbner bases
Nikolay N. Vassiliev, Dmitry Pavlov · Programming and Computer Software · 2009
The goal of this work is to analyze various classes of finite and total monomial orderings. The concept of monomial ordering plays the key role in the theory of Gröbner bases: every basis is determined by a certain ordering. At the same time, in order to define a Gröbner basis, it is not necessary to know ordering of all monomials. Instead, it is sufficient to know only a finite interval of the given ordering. We consider combinatorics of finite monomial orderings and its relationship with cells of a universal Gröbner basis. For every considered class of orderings (weakly admissible, convex, and admissible), an algorithm for enumerating finite orderings is discussed and combinatorial integer sequences are obtained. An algorithm for computing all minimal finite orderings for an arbitrary Gröbner basis that completely determine this basis is presented. The paper presents also an algorithm for computing an extended universal Gröbner basis for an arbitrary zero-dimensional ideal.